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Photon-assisted electron transport through a three-terminal quantum dot system withnonresonant tunneling channels

T. Kwapiński,* R. Taranko,† and E. Taranko‡

Institute of Physics, Maria Curie-Skłodowska University and Maria Curie-Skłodowska University Nanotechnology Center,20-031 Lublin, Poland

�Received 2 March 2005; revised manuscript received 12 July 2005; published 8 September 2005�

We have studied the electron transport through a quantum dot coupled to three leads in the presence ofexternal microwave fields supplied to different parts of the considered mesoscopic system. Additionally, weintroduced a possible nonresonant tunneling channel between leads. The quantum dot charge and currents weredetermined in terms of the appropriate evolution operator matrix elements and under the wide-band limit theanalytical formulas for time-averaged currents and differential conductance were obtained. We have alsoexamined the response of the considered system on the rectangular-pulse modulation imposed on differentquantum dot-lead barriers as well as the time dependence of currents flowing in response to suddenly removed�or included� connection of a quantum dot with one of the leads.

DOI: 10.1103/PhysRevB.72.125312 PACS number�s�: 73.23.�b, 73.63.Kv

I. INTRODUCTION

The electron transport via resonant tunneling in mesos-copic systems has been the subject of extensive theoreticalresearch due to recent development in fabrication of smallelectronic devices and their potential applications. Some in-terest has been focused on the transport properties of a quan-tum dot �QD� under the influence of external time-dependentfields. Interesting effects have been observed and theoreti-cally described, e.g., photon-assisted tunneling through smallquantum dots,1,2 photon-electron pumps3 and others. In mosttheoretical investigations a QD placed between two leadswas considered �e.g., Refs. 2 and 4–9� and the current flow-ing through a QD under periodic modulation of the QD elec-tronic structure or periodic �nonperiodic� modulation of thetunneling barriers and electron energy levels of both �left andright� leads was calculated.

One of the important problems of the mesoscopic physicsis the interference of the charge carries. This interferenceappears when two �or more� transmission channels for elec-tron tunneling exist. Such possibility exists, e.g., in the elec-tron transport through a QD embedded in a ring in theAharonov-Bohm geometry and much theoretical interest hasbeen paid to description of the phase coherence in this andrelated systems, e.g., Refs. 10–12. Another experimentalsituation in which the interference may occur can be realizedwith the scanning tunneling microscope �STM�. The recentexperimental and theoretical studies with a low-temperatureSTM of a single atom deposited on a metallic surfaceshowed the asymmetric Fano resonances in the tunnelingspectra, e.g., Refs. 13 and 14. In the STM measurements thetip probes the transmission of electrons either through theadsorbed atom or directly from the surface. The transport ofelectrons through both channels leads to an asymmetricshape of the conductance curves which is typical behaviorfor the Fano resonance resulting from constructive and de-structive interference processes. The quantum interferencecan be also observed in the mesoscopic system with multipleenergy levels.15 A model which incorporates a weak direct

nonresonant transmission through a QD as well as the reso-nant tunneling channel was also considered in Ref. 16 in thecontext of the large value of the transmission phase found inthe experiment for the Kondo regime of a QD.17

A number of works have been devoted to the problem ofthe electron transport in the multiterminal QD systems andhere we mention only a few of them. In Ref. 18 the conduc-tance of the N-lead system was considered showing that theKondo resonance at equilibrium is split into N−1 peaks. InRef. 19 an explicit form for the transmission coefficient inthe electron transport through a QD connected with threeleads was found. The electron transport and shot noise in amultiterminal coupled QD system in which each lead wasdisturbed by classical microwave fields were studied in Ref.20. Multiterminal QD systems or magnetic junctions werealso intensively investigated in context of the spin-dependenttransport, e.g., Refs. 21 and 22. The general formulation ofthe time-dependent spin-polarized transport in a system con-sisting of the resonant tunneling structure coupled with sev-eral magnetic terminals was considered by Zhu et al.22 andas an application of this formalism the electron transport in asystem with two terminals under an ac external field wasinvestigated. A three-terminal QD system was studied inRefs. 23 and 24 to measure the nonequilibrium QD densityof states �splitting of the Kondo resonance peak�. The crosscorrelations of the currents and the differential conductanceof the QD coupled with three leads described by the infinite-U Anderson Hamiltonian were considered in Ref. 25; seealso Ref. 26. A mesoscopic sample connected to many res-ervoirs via single channel leads was considered in Ref. 27 inthe context of the quantum pump effect. The transport prop-erties of molecule wires or coupled quantum dots placed be-tween three leads and irradiated by the infrared light quantawere studied in Ref. 28 and the possibility of controlling thecurrents flowing in the system was discussed.

In all papers mentioned above and relating to the studiesof the electron transport through a two-terminal QD with theadditional �bridge� nonresonant transmission channel, thetime-dependent external fields were not applied and the con-

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sidered systems were driven out of the equilibrium only bymeans of a dc voltage bias �see, however, Ref. 29�. On theother hand, in papers considering the multiterminal QD sys-tems in the presence of the external microwave fields, thenonresonant tunneling channels between different leads werenot considered.

In this paper, we address the issue of a QD coupled withthree leads with additional, nonresonant coupling betweenleads driven out of equilibrium by means of a dc voltage biasand time-dependent external fields. The QD is connectedwith three metal leads and one of these leads, say the left �L�lead, is coupled with the two remaining leads, say the first�R1� and the second �R2� right leads. The one possible ex-perimental setup �e.g., in the STM measurement� corre-sponding to our model system is presented in Fig. 1. We havedecided to denote the three leads by the subscripts L, R1, andR2 in order to have the simplified comparison with the two-terminal QD systems described in literature �usually pre-sented as left lead–QD–right lead�. In literature, differenttheoretical approaches have been developed to treat the time-dependent electron transport in the mesoscopic systems. Themost popular nonequilibrium Green’s-function method de-pends on the two time arguments and for time-dependentproblems it is a rather difficult task to calculate them withoutany approximations �e.g., beyond the wide-band limit�.Therefore in our treatment of the time-dependent problemswe use the evolution operator which, as a rule, essentiallydepends on one time argument �e.g., Refs. 30–32�. Such anapproach is especially well suited for the problem with time-dependent coupling between the QD and leads, see also Ref.33 where the evolution operator method was used in a de-tailed investigation of the zero-frequency noise and time-dependent current in a mesoscopic system irradiated by thelaser radiation or microwave field.

The outline of the paper is as follows. In Secs. II and IIIwe start with the model and method for the derivation of theQD charge and currents. In Sec. IV we present the results forthe time-averaged currents and their derivatives with respectto the QD energy level position �or equivalently, with respectto the gate voltage� obtained for different time dependence ofthe parameters characterizing the considered system. Weconsider also the transient current characteristics in the caseof the rectangular-pulse modulations imposed on the QD-

lead barriers. The last section presents the conclusions and inthe Appendix we give the short derivation of the evolutionoperator matrix elements needed in the QD charge and cur-rent calculations.

II. MODEL AND GENERAL FORMULATION

We consider a QD coupled through the tunneling barriersVk�id

�i=1,2,3� to three metal leads. One of these leads, saythe left lead �L� is coupled additionally with the remainingtwo leads, say the first and second right leads �R1 ,R2� by thetunneling barriers Vk�L,k�R

. In the following we will denote thewave vectors associated with the left lead by the letter k� andthe wave vectors corresponding to the first and second rightleads by the letters q� and r�, respectively. The chemical po-tentials �i of the three metal leads may not be equal, andtheir difference is not necessarily small. We write the Hamil-tonian of the considered system in the form H=H0+V, where

H0 = �p�=k�,r�,q�

�p��t�ap�+ap� + �d�t�ad

+ad, �1�

V = �p�=k�,r�,q�

�Vp�d�t�ap�+ad + H . c . � + �

k�,r�

�Vk�r��t�ak�+ar� + H . c . �

+ �k�,q�

�Vk�q��t�ak�+aq� + H . c . � . �2�

The operators ap��ap�+� , ad�ad

+� are the annihilation �creation�operators of the electrons in the corresponding leads and thedot, respectively. For simplicity the dot is characterized onlyby a single level �d and the intradot electron-electron Cou-lomb interaction is neglected. The neglect of Coulomb inter-action is quite reasonable in some situations and, as we aregoing to concentrate on the investigations of the influence ofthe third lead �in comparison with the QD–two leads system�and the additional tunneling channels between the leads onthe time-dependent transport, then in the first step we ignorethe Coulomb interaction. We obtain, for example, as the oneof the consequences of neglecting of intradot Coulomb inter-action, the average current vs the QD energy level in theform of a single peak instead of a series of oscillation peaks.However, as the experiment indicates, e.g., Ref. 3, each ofthese peaks is modified in a similar manner by the externaltime-dependent field, so our analysis of one of these peaksshould be justified. We consider our mesoscopic system inthe presence of external microwave fields which are appliedto the dot and three leads. In most theoretical treatments ofphoton-assisted electron tunneling it is assumed that the driv-ing field equals the applied external field. However, the situ-ation is more complicated and the internal potential can bedifferent from the applied potential. One of the consequenceswill be, e.g., the asymmetry between the corresponding leftand right sidebands.34 The main feature of the time-dependent transport remains, however, unchanged and in ourtreatment as usual we assume that in the adiabatic approxi-mation the energy levels of the leads and QD are driven withthe frequency � and the amplitudes �i�i=L ,R1 ,R2� , �d andread �k�i

�t�=�k�i+�icos �t , �d�t�=�d+�dcos �t, respectively.

FIG. 1. Schematic picture of the QD coupled with three leads. Itcan serve as a possible STM experimental setup when the L leadcorresponds to the STM tip.

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The time evolution of the QD charge and the currentflowing in the system can be described in terms of the time-evolution operator U�t , t0� given by the equation of motion�in the interaction representation�

i�U�t,t0�

�t= V�t�U�t,t0� , �3�

with V�t�=U0�t , t0�V�t�U0+�t , t0� and U0�t , t0�

=T exp�i�t0t dt1H0�t1�� where T denotes the time ordering and

the units such that �=1 have been chosen. Here we haveassumed that the interactions between the QD and leads, aswell as between the left and right leads, are switched on inthe distant past t0.

The QD charge �in units of e� is calculated according tothe formula �cf. Refs. 30 and 31�

nd�t� = nd�t0��Udd�t,t0��2 + �p�=k�,r�,q�

np��t0��Udp��t,t0��2. �4�

Here Udd�t , t0� and Udp��t , t0� denote the matrix elements ofthe evolution operator calculated within the basis functionscontaining the single-particle functions �k��, �q��, �r��, and �d�corresponding to three leads and QD, respectively. nd�t0� andnp��t0� represent the initial filling of the corresponding single-particle states.

The tunneling current flowing, e.g., from the left leadjL�t�, can be obtained using the time derivative of the totalnumber of electrons in the left lead, jL�t�=−ednL�t� /dt,where

nL�t� = �k�

nk��t� = �k�

nd�t0��Uk�d�t,t0��2 + �k�,k�1

nk�1�t0��Uk�k�1

�t,t0��2

+ �k�,q�

nq��t0��Uk�q��t,t0��2 + �k�,r�

nr��t0��Uk�r��t,t0��2. �5�

In the following only the matrix elements of the evolutionoperator present in Eqs. �4� and �5� are required and they canbe obtained solving the corresponding sets of coupled differ-

ential equations constructed according to Eq. �3� with Vab�t�written as follows:

Vab�t� = Vab�t�expi��a − �b��t − t0�

+ i�a − �b

��sin �t − sin �t0� , �6�

where a and b correspond to �k��, q��, �r��, or �d�, respectively.As the example, the matrix element Udd�t , t0� required for

the calculation of the first term of the QD charge, Eq. �4�,can be obtained solving the following set of coupled differ-ential equations:

�Udd�t,t0��t

= − i �p�=k�,q� ,r�

Vdp��t�Up�d�t,t0� , �7�

�Uk�d�t,t0�

�t= − iVk�d�t�Udd�t,t0� − i �

p�=q� ,r�Vk�p��t�Up�d�t,t0� ,

�8�

�Up�d�t,t0�

�t= − iVp�d�t�Udd�t,t0� − i�

k�Vp�k��t�Uk�d�t,t0�,

p� = q� ,r� . �9�

The total number of coupled equations in this case isequal to �3N+1� , N being the number of discrete wave vec-tors k�, q� , and r� taken to perform the corresponding summa-tion over the wave vectors. In the calculations the summa-tions over wave vectors were replaced, as usual, withintegrations over energy weighted with a given lead densityof states �taken as a constant value�. Usually, N=100–200wave vectors, or equivalently 100–200 energy points uni-formly distributed along the lead energy band, is sufficient toachieve the desired accuracy of the calculations. We havesolved numerically this and other similar sets of the coupleddifferential equations needed in calculations of all matrix el-ements of the evolution operator present in Eqs. �4� and �5�.We have used this method for the special case of time-dependent couplings of the QD with leads and the couplingsof the left lead with two right leads. The set of Eqs. �7�–�9�in the case of vanishing overdot couplings between the leftand right leads is greatly simplified and gives, e.g., forUdd�t , t0�,


= − �t0

t

dt1K�t,t1�Udd�t,t0� , �10�

where

K�t,t1� = �p�=k�,q� ,r�

Vdp��t�Vp�d�t1�

= �i=L,R1,R2

�Vi�2ui�t�ui�t1�Di�t − t1�exp�i�d�t − t1��

�exp�i��d − �i��sin �t − sin �t1�/�� 11�

and Di�t− t1� is the Fourier transform of the ith lead densityof states and Vdi�t�=Viui�t�. Similar simplifications occur inthe calculations of other matrix elements of U�t , t0� requiredin the formulas for nd�t� and nL�t�. However, for the nonva-nishing couplings Vk�q� and Vk�r� �overdot bridge between theleft and right leads� one has to solve the starting set of Eqs.�7�–�9�.

III. ELECTRON TRANSPORT IN THE WIDE-BAND LIMIT

Much more analytical calculations can be done in the caseof constant values of the tunneling matrix elements presentin our model. In this case the general equation satisfied byUdd�t , t0� is derived in the Appendix �see Eq. �A5�� and underthe wide-band limit �WBL� approximation, e.g., Refs. 2, 4,and 5, this equation takes the simple form

PHOTON-ASSISTED ELECTRON TRANSPORT THROUGH … PHYSICAL REVIEW B 72, 125312 �2005�

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= − C1Udd�t,t0� , �12�

here C1=� 32 − �3x2+2ix� / �1+2x2�� , x=VRL /D , D being

the bandwidth of the lead energy band �DR1=DR2

=DL=D�and �=2V2 /D. In the Appendix we give the derivations ofall functions needed for calculation of the QD charge andcurrents. We assumed the simplified assumption that all tun-neling matrix elements are independent of the wave vectors.The interactions between the QD and leads are assumed to beequal between themselves and denoted by V and the interac-tions between the left and two rights leads �i.e., Vk�q� and Vk�r��corresponding to the overdot tunneling channels are alsoequal with one another and denoted by VLR.

It is easy to show that the first term of the general formulafor the QD charge, Eq. �4�, together with the solution of Eq.�12�, Udd�t , t0�=exp�−C1�t− t0��, tends to zero for t− t0→as Re C1=3� /2−3x2� / �1+2x2��0. The next terms of theQD charge formula can be calculated using the functionsUdk��t , t0�, Udq��t , t0�, and Udr��t , t0�, Eqs. �A13� and �A17� be-ing the solutions of the corresponding differential equations,Eqs. �A12� and �A16�. Finally, the time-averaged QD charge�in units of e� is given by

�nd�t�� = �i=L,R1,R2

ai� d�f i��Ai��,t��2� , �13�

where

aL = �1 + 4x2�/�1 + 2x2�2�/2 , �14�

aR1= aR2

= �1 + x2�/�1 + 2x2�2�/2 , �15�

Ai��,t� = − i�t0

t

dt1exp − i��d − ��t − t1� − i��d − �i�

��sin �t − sin �t1�/��exp��− 3 + i4x�2�1 + 2x2�

�t − t1� .

�16�

Here �¯� denotes the time averaging and f i�� denotes theFermi function of the ith �i=L ,R1 ,R2� lead. Noticing thatIm�Ai�� , t��=−�3� /2 / �1+2x2��Ai�� , t��2� �cf. Ref. 5�, theexpression for the time-averaged QD charge can be writtenas

�nd�t�� = − Im 1 + 4x2

3�1 + 2x2� � fL��AL��,t��d�

+1 + x2

3�1 + 2x2� �i=1,2

� fRi��ARi

��,t��d� .

�17�

In order to calculate the current jL�t� the functions Uk�d�t , t0�,Uk�1k�2

�t , t0�, Uk�q��t , t0�, and Uk�r��t , t0� are required and they aregiven in the Appendix in Eqs. �A7�, �A14�, and �A19�. Afterlengthly but straightforward calculations we obtain for thetime-averaged current leaving the left lead the following for-mula:

�jL�t�� =e

��

i=R1,R2

Re 2x2

�1 + 2x2�2 ��L − �i�

+ G� fL��AL��,t��d� −� f i��Ai��,t��d�� ,

�18�

where

G =�

3�1 + 2x2�3 �6x�1 − 2x2� + i�1 − 13x2 + 4x4�� ,

�19�

�Aj��,t�� = �k

Jk2�d − � j

�

�� − �d − �k +2�x

1 + 2x2 + i3�/2

1 + 2x2−1

,

�20�

and Jk�y� denotes the Bessel function. The correspondingformula for the time-averaged current �jRi

�t�� leaving the Ri

lead, i=1,2, cannot be written in such symmetrical form as inEq. �18�, because the Ri lead is coupled with the L lead only.For �jRl

�t�� we have

�jRl�t�� =

e

�Re 2x2

�1 + x2�2 ��Rl− �L + x2��Rl

− �Rj��

+�

3�1 + 2x2�32G2� fRl��ARl

��,t��d�

− G1� fL��AL��,t��d� − G3� fRj��

��ARj��,t��d�� , �21�

where G1=6x�1−2x2�+ i�1−13x2+4x4�, G2=3x− �i /2��−2+5x2+x4�, G3=12x3+ i�1+8x2−5x4� and j=1�2� for l=2�1�.Note that the integrals present in the formula for the QDcharge and currents, Eqs. �17�, �18�, and �21� can be easilyperformed analytically and final algebraic expressions can beobtained. Especially simple and transparent form can begiven for the conductance �� /��i��j j�t�� , i , j=L ,R1 ,R2. Forexample, ��jL�t�� /��L reads as �for the temperature T=0�

�

��L�jL�t�� =

e

� 4x2

�1 + 2x2�2 + �k

Jk2�d − �L

�

F1�2�1 − 13x2 + 4x4��1 + 2x2�4

+4�x�1 − 2x2��1 + 2x2�3 F2� , �22�

where


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F1 = �L − �d − �k +2�x

1 + 2x22

+ 3�/2

1 + 2x22�−1

,

�23�

F2 = �L − �d − �k +2�x

1 + 2x2 . �24�

Analyzing Eq. �22� one can find the origin of the asymmetricline shapes in differential conductance resulting from the in-terference of resonant and nonresonant tunneling paths. Forthe case of VLR=0 we observe the Lorentzian resonanceslocalized at �d=�L±�k. The amplitudes of these resonancesare determined by the kth-order Bessel functions calculatedfor the argument ��d−�i� /�. For the case of nonvanishingVLR, the resulting curve is a superposition of the Lorentzian-like resonances and asymmetric parts weighed by the factors�2�1−13x2+4x4� /�1+2x2�4 and 4�x�1−2x2� /�1+2x2�3,respectively. The Lorentzian-like resonance is centeredat �d=�L±�k+2�x / �1+2x2� with the full width at halfmaximum �FWHM� equal to 3� / �1+2x2� and the maximumvalue equal to �4/9��1−13x2+4x4� / �1+2x2�2�Jk

2��d

−�L� /��. The asymmetric part of the differential conduc-tance corresponding to the kth sideband is also centered inthe same point with the distance between its maximum andminimum equal to 3� / �1+2x2� and the absolute valuesof these extrema are equal to �4/3��x�1−2x2� / �1+2x2�2�Jk

2��d−�L� /��. For comparison, in the case of theQD coupled with two leads the corresponding Lorentzian-like part of the differential conductance corresponding to thekth sideband is centered at �d=�L±�k+�x / �1+x2�, withFWHM equal to 2� / �1+x2� and the maximum value equal to�1/2��1−6x2+x4� / �1+2x2�2�Jk

2��d−�L� /��. Knowingthe explicit expressions for the currents one can check thefollowing relations between different elements of the con-ductance matrix −e� �jn�t�� /��m, e.g., Refs. 18 and 26. Cur-rent conservation implies �n� �jn�t�� /��m=0 where n ,m=L ,R1 ,R2. On the other hand, �m� �jn�t�� /��m=0 only for�d−�L=�d−�R1

=�d−�R2. For other relations between the

amplitudes �d and �L ,�R1,�R2

we have

�k

� �jR1�t��/��k = �

k

� �jR2�t��/��k = −

1

2�k

� �jL�t��/��k

�25�

for �d−�R1=�d−�R2

��d−�L, and

�k

� �jR1�t��/��k � �

k

� �jR2�t��/��k � �

k

� �jL�t��/��k

�26�

for �d−�R1��d−�R2

��d−�L.

IV. RESULTS AND DISCUSSION

We consider the QD coupled with three, say the left, thefirst right, and the second right, metal leads with the addi-tional overdot �bridge� couplings between the left and right

leads. First, in the Sec. IV A and IV B, we present the resultsfor the time-averaged currents and derivatives of the averagecurrent with respect to the QD energy level in the presenceof external microwave fields which are applied to the dot andthree leads, respectively. Assuming the harmonic time modu-lation of the external fields, here we give the explicit formulafor the averaged current, �jL�t��, performing the correspond-ing integrals in the general formula given in Eq. �18�, andcompare it with the current �jL

�2��t�� flowing in the system ofthe QD coupled with two leads only �for the zero-temperature case�:

�jL�t�� =e

��

i=R1,R2

� 2x2

�1 + 2x2�2 ��L − �i� +�

3

1 − 13x2 + 4x4

�1 + 2x2�3

� �k Jk

2�d − �L

�arctan�hL

�k��

− Jk2�d − �i

�arctan�hi

�k��+

�

x�1 − 2x2��1 + 2x2�3 �

k Jk

2�d − �L

�ln�gL

�k��

− Jk2�d − �i

�ln�gi

�k�� , �27�

where hi�k�= ��i−�d−�k+2�x / �1+2x2�� / �3� /2�1+2x2��,

gi�k�= ��i−�d−�k+2�x / �1+2x2��2+ �3� /2�1+2x2��2, and i

=L ,R1 ,R2, whereas for �jL�2��t�� we have

�jL�2��t�� =

e

�� 2x2

�1 + x2�2 ��L − �R� +�

2

1 − 6x2 + x4

�1 + 2x2�3

� �k Jk

2�d − �L

�arctan�hL

�2k��

− Jk2�d − �R

�arctan�hR

�2k��+

�

x�1 − x2��1 + x2�3 �

k Jk

2�d − �L

�ln�gL

�2k��

− Jk2�d − �i

�ln�gR

�2k�� , �28�

where hi�2k�= ��i−�d−�k+�x / �1+2x2�� / �� / �1+2x2�� and

gi�2k�= ��i−�d−�k+�x / �1+x2��2+�2 / �1+x2�2 , i=L ,R. Note

that �jL�t�� consists of the two terms and each term is similarin its structure to the current flowing in the QD–two leads�QD-2l� system, �jL

�2��t��. However, due to the interference ofthe charge carriers propagating along the different ways, thearguments of the inverse tangent and logarithmic functionsare different and the individual terms in �jL�t�� and �jL

�2��t��are weighted by different x-dependent factors. Second, inSec. IV C, the time-dependent currents are also calculated inthe case when the periodic or nonperiodic rectangular-pulseexternal fields are applied to each QD-lead barrier.


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We have taken for VLR the values comparable with Vk��d

and estimated Vk��d �assuming its k� independence, Vk��d�V�

=V� using the relation ��=2�V��2 /D�, where D� is the�-lead bandwidth and D�=100 �� L=�R=� , DL=DR=Dwas assumed�. We also assumed the zero-temperature case,as the temperature in the experiment is usually low. For cal-culations, we have chosen e=�=1 units and all energies aregiven in the � units.

A. Microwave field applied to the QD only

In Fig. 2 we compare the averaged values of the currentflowing from the left lead in the systems in which the QD iscoupled with three or two leads, the solid and broken curves,respectively. The external microwave field is applied only tothe QD and dc bias between the left and right leads is smallin comparison with �, �d, and �. The coupling VLR is as-sumed to be zero. In such a case the sidebands on the currentcurve are clearly visible. The width of the correspondingpeaks is smaller for the case of the QD coupled with twoleads. Analyzing the expression for �jL�t�� one can obtain�for �L � and �R1

=�R2=0� the subsequent peaks in �jL�t��

�as functions of �d� in the form �4/9�Jk2��d /��L�1+ ��d

+�k� /3� /2�2�−1 with the FWHM equal to 3�. For compari-son, in the case of the QD coupled with two leads the corre-sponding peaks are described by the functions�1/2�Jk

2��d /��L�1+ ��d+�k� /��2�−1 with the FWHMequal to 2�.

Next, we consider the case of a QD coupled with twoleads with varying value of �R and compare it with the QDcoupled with three leads �with the chemical potential of thethird lead �R2

=−�L�.In Fig. 3 we present the results obtained for �jL�t�� for the

QD coupled with three �thick lines� or two �thin lines� leads.The upper �lower� panel corresponds to x=0�x=0.28�. Forthe case �L=−�R1

=−�R2=0.1 and x=0 the sidebands are

very clearly visible and the corresponding peaks are lowerand broader for the QD–three leads �QD-3l� system as we

discussed before �the upper panel, the solid lines�. For thenonzero overdot coupling between the left and right leads thesideband peaks get asymmetric forms and for x=0.28 theybecome fully asymmetric �lower panel, solid lines�. The cur-rent for the QD-2l system is also composed of a number ofasymmetric components �thin solid lines� although now theseforms are fully asymmetric for x=0.41 as we know from theearlier discussion. For greater values of �R1

, the correspond-ing current flowing in the QD-2l system achieves �for van-ishing, as well as for nonzero coupling between leads�greater negative values and its dependence on the QDenergy-level position is well marked in comparison to theresults characterizing the QD-3l system.

Next, in Fig. 4 we analyze the influence of the overdotcoupling between the left and right leads for broad range ofthis coupling. The thin �thick� lines correspond to the QDcoupled with two �three� leads. We show �jL�t�� for threevalues of the interleads coupling strength represented by the

FIG. 2. The averaged current �jL�t�� against the QD energy level�d in the system of a QD coupled with three �solid line� or two�broken line� leads. The microwave field is applied to the QD withthe amplitude �d=6 and frequency �=5. The lead chemical poten-tials �L, �R1

, and �R2are equal 0.2, 0.0, and 0.0, respectively, and

the coupling between L and R1 , R2 leads is absent, VLR=0. Allenergies are in � units and the current in e� /� units.

FIG. 3. The averaged current �jL�t�� against �d in the system ofa QD coupled with three �thick lines� or two �thin lines� leads. Theupper �lower� panel corresponds to x=0 �x=0.28� , x being the mea-sure of the coupling between leads, x=VLR /D. In a QD–threeleads system �L=0.1, �R2

=−0.1, and �R1=−0.1, 4, and 10 �solid,

broken, and dotted curves, respectively�. In a QD coupled with twoleads �L=0.1, �R=−0.1, 4, and 10 �solid, broken, and dottedcurves, respectively�. �=5, �d=8, �R1

=�R2=�L=0. The values of

solid curves have been multiplied by a factor of 10 for the illustrat-ing purposes.


125312-6

parameter x=VLR /D. For x=0 we have well defined side-band peaks as previously shown in Fig. 2. Next, we show theresults for x=��13−�153� /8�0.28 and x=�2/2�0.7. Notethat the expression for �jL�t�� and �jL

�2��t�� consists of threeterms. The first term depends on the difference �L−�i anddoes not depend on the QD energy level �d. The second termcorresponds to the Loretzian-type contribution to a givensideband �it disappears for x�0.28� and the last term corre-sponds to an asymmetric contribution �it disappears for x�0.7�, respectively. This last term influences the sidebandshape only for nonvanishing overdot coupling between leadsand is the most prominent sign of the interference effects.For x=�2/2 this term disappears and the resulting sidebandshave a form of symmetric dips due to the negative value ofthe coefficient ��1−13x2+x4� / �3�1+2x2�3� in Eq. �27�. Inthis case the nonresonant tunneling channels modify thedip’s center position and its FWHM in comparison with theposition and FWHM of sidebands presented for x=0. Notethat for a QD coupled with two leads, the correspondingsidebands �the thin dotted line� are not fully symmetriccurves as in this case the last term of Eq. �28� disappears forx=1 and not for x=0.7 �cf. Ref. 29�. For x�0.28 the corre-sponding sidebands are fully asymmetric curves as in thiscase the second term in Eq. �27� �which introduces asymme-try� disappears. Again, for a QD coupled with two leads thecorresponding sidebands �the thin broken curve� are de-scribed by not fully asymmetric curves as the second term ofEq. �28�, which gives a symmetric contribution to sidebands,disappears for x=�3−2�2�0.41 and not for x=0.28.

In order to emphasize the influence of the additional leadon the currents and find the possible interference effects, weshow in Fig. 5 the current �jL�t�� for QD-2l and QD-3l sys-tems calculated for the parameters for which the correspond

ing curves are relatively simple. We assumed the small am-plitude of the QD energy level oscillations, �d=1, and �=5, for which �for x=0� only the central peak correspondingto elastic tunneling is visible on the �jL�t�� curves. We show�jL�t�� obtained for two different QD-2l systems which canbe viewed as components of the considered more compli-cated QD-3l system. We observe that due to the interferenceeffects, the current �jL�t�� flowing in the QD-3l system �thebroken lines� is not simply a sum of currents �dotted lines�flowing in the corresponding QD-2l systems. The differencebetween this sum and the current corresponding to the QD-3lsystem is relatively large and exists independently of thecoupling between leads.

In the next Fig. 6 we show all currents, �jL�t��, �jR1�t��,

and �jR2�t��, flowing in the QD-3l system for x=0 and x

=0.28. For vanishing values of the overdot coupling the cur-rent �jL�t�� is characterized by a sequence of the symmetricpeaks, but the current �jR2

�t�� is a superposition of the asym-metric structures placed in the points where the symmetricsidebands occur on �jL�t�� curve. Analyzing the current�jR2

�t�� according to Eq. �21� we have for the parameters inFig. 6.

FIG. 4. The averaged current �jL�t�� against �d in the system ofa QD coupled with three �thick lines� or two �thin lines� leads. Thesolid, broken, and dotted curves correspond to x=0, 0.28, and 0.7,respectively. �d=8, �R1

=�R2=�L=0, �L=−�R1

=−�R2=0.1,

�=5.

FIG. 5. The averaged current �jL�t�� against �d for the QD-2lsystems: �L=−�R=0.1—thin solid lines and�L=0.1, �R=4—thick solid lines and the QD-3l system: �L=−�R1

=0.1, �R2=4—broken lines. The upper �lower� part corre-

sponds to x=0 �x=0.28� and �=5, �d=1, �R1=�R2

=�L=0. Thedotted lines correspond to the sum of the currents flowing in twoQD-2l systems.


125312-7

�jR2�t�� =

e

� �

3�

k

Jk2�d

��arctan�hR2

�k�� − arctan�hL�k��

+ arctan�hR2

�k�� − arctan�hR1

�k�� , �29�

where hi�k�= ��i−�d−�k� / �3� /2�. One can see that each

sideband is the sum of the peak �two first terms in Eq. �29��and the dip �the last two terms in Eq. �29�� resulting in theasymmetric structure shown in Fig. 6

To learn more about the influence of the third electrodeand additional overdot coupling between leads we present inFig. 7 the currents �jL�t�� for x=0 and x=0.28 and �jR2

�t�� forx=0 as functions of the QD level position �d and the fre-quency � of the microwave field applied to the QD. Forvanishing coupling between leads �x=0� the current �jL�t��exhibits a well-known sideband structure for �� and forsmall frequencies, ��, the two broad maxima at �d= ±�d are present. At the same time, the current �jR2

�t�� ex-hibits the asymmetric structures centered on the ��d ,�� planeat the points where photon sidebands occur on the �jL�t��curves. These asymmetric structures on the �jR2

�t�� curve ex-ist also at ��. On the other hand, a similar structure of the�jL�t�� the lower panel of Fig. 7� is obtained for x=0.28, i.e.,

we observe a number of asymmetric resonances separated bythe photon energy for ��. Notice the similarity of bothpictures, i.e., �jL�t�� calculated for x=0.28 and �jR2

�t�� for x=0, respectively. Note, however, the different scale for thesecurrents.

B. Microwave field applied to different parts of the system

In Fig. 8 we present �jL�t�� for different overdot couplingassuming a strong asymmetry of the applied microwave field�ac potential is applied only to the R1 lead in the QD-3lsystem�. The additional R2 lead is characterized by thechemical potential �R2

= ��L+�R1� /2=0. For better demon-

FIG. 6. The averaged currents �jL�t��, �jR1�t��, and �jR2

�t��solid, broken, and dotted lines, respectively� against �d in the sys-tem of a QD coupled with three leads. The upper �lower� panelcorresponds to x=0�x=0.28�. �d=8 and other parameters are as inFig. 4. The values of �jR2

�t�� for x=0 �upper panel� have beenmultiplied by a factor of 20 for illustrating purposes.

FIG. 7. The averaged currents against �d and �. The upper�middle� panel shows �jL�t��jR2

�t�� for x=0 and the lower panelgives �jL�t�� for x=0.28. �d=8, �R1

=�R2=�L=0, �L=−�R1

=−�R2

=0.2.


125312-8

stration of the influence of the ac potential on �jL�t�� wemoved down each curve by the constant value A= �e /��2x2�2�L−�R1

−�R2� / �1+2x2�2; see Eq. �27�. This

constant value does not depend on the ac field and is equal tothe current obtained for the static case in the limit of largevalues of �d �the current between leads is practically inducedby the overdot tunneling, only�. For x=0 we observe a shoul-der on the left side of the main peak and a small negativecurrent for the positive values of �d. This picture is similar tothe known results �for x=0� obtained experimentally andtheoretically in the QD-2l systems, e.g., Refs. 3 and 35. Withthe increasing overdot coupling VLR the height of the mainresonant peak decreases and disappears for x=0.28. At thesame time, for all values of x we observe a negative currentfor the small values of �d with a greater absolute value forstronger coupling between leads. For greater values of x theshape of the �jL�t�� curve is changed dramatically and for x=0.7, �jL�t�� becomes nearly reversed in comparison withthat calculated for x=0.

In Fig. 9 we analyze the influence of the third lead �herenamed as R2� on the current �jL�t�� when the external micro-wave field is applied to this lead and to the QD with �d=2�R2

. For comparison, we add in the upper panel the resultsfor �jL�t�� obtained for the case when this additional thirdlead is not irradiated by the microwave field. In this case, asbefore, see, e.g., Fig. 2, we observe typical sidebands on thecurrent curves �the difference between the lead chemical po-tentials is small in comparison with the amplitude �d�. How-ever, after including the third lead irradiated by the externalmicrowave field the dependence of the current �jL�t�� jL onthe gate voltage �or equivalently on the QD energy levelposition� is quite different—compare the thin or thick solidlines of the upper and lower panels. For smaller values of �dand �R2

, the averaged current jL is very similar to the corre-sponding current JL obtained by applying the external micro-wave field only to one lead �see the thick solid line in Fig. 8�.These curves are, however, related between themselves by arelation JL��d�� jL�−�d�. Now we can observe a small nega-

tive current at small negative values of �d and some enhance-ment of the current on the right side of the main peak. Simi-larly, the significant differences between the correspondingcurrents are observed also for greater values of the ampli-tudes �d and �R2

�compare the thick solid lines in the upperand lower panels of Fig. 9�. Note that very similar behaviorof the current �jL�t�� as the function of the gate voltage isobserved if we compare the case when the microwave field isapplied only to one lead and the case when the microwavefield is applied simultaneously to the QD and R2 lead butwith the phase difference of —compare the thin brokencurve in Fig. 9 with the thick solid curve in Fig. 8.

In order to present more information about the differencesbetween the electron transport in the QD-3l and QD-2l sys-tems we display in Fig. 10 the derivative d�jL�t�� /d�d as afunction of �d and �. The lowest panel corresponds to theQD-3l system and two other panels correspond to the QD-2lsystems. These QD-2l systems are characterized by such pa-rameters that combined together give us the consideredQD-3l system. One can see that the considered characteris-tics of the electron transport in the QD-3l system are notsimply the algebraic sum of the corresponding curves of bothQD-2l systems. In all three cases shown in Fig. 10, the po-sition of the corresponding minima and maxima �along the�d axis� can be identified with the values of the leads chemi-cal potentials. However, the corresponding structures are lessclear in the case of the QD-3l system in comparison withthose for the QD-2l models.

FIG. 8. The averaged current �jL�t�� against �d for differentvalues of the overdot coupling strength between the left and rightleads. The thick solid, solid, dotted, and broken lines correspond tox=0, 0.14, 0.28, and 0.7, respectively. A= �e /��2x2�2�L−�R1−�R2

� / �1+2x2�2 and �R1=3 , �L=�R2

=�d=0, �L=−�R1=0.2, �R2

=0 , �=5.

FIG. 9. The averaged current �jL�t�� against �d for the case ofthe microwave field applied only to the QD with �d=3 �thin line� or�d=6 �thick line�—upper panel. The lower panel corresponds to thecase of the microwave field applied to the QD and R2 lead with�d=2�R2

and �R2=1.5 or 3 �thin or thick lines, respectively�. The

broken lines show the results when the microwave field applied tothe QD and R2 lead are out of phase �with the phase difference of�, �L=−�R1

=0.2, �R2=0 , �=5.


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C. Rectangular-pulse modulation of the couplings between QDand leads

In the next step of our investigations of the electron trans-port in the QD-3l systems we consider the time-dependentcurrents flowing in response to the time-dependent barriersbetween the QD and leads or in response to suddenly re-moved �or inserted� connection of the QD with one of theleads. First we assume a rectangular-pulse modulation ap-plied to the QD-R1 lead and QD–R2 lead barriers. We assume

that these modulations are with a phase difference of . Inthe first �second� half cycle, VdR1

=0�VdR2=0� and the QD is

coupled only to the R2 lead �R1 lead�. In addition, the QD iscoupled to the next, say L lead, with a constant value VdL. Inthe following we consider the time-dependent currents jL�t�,jR1

�t�, and jR2�t� for the three specific conditions: �L=�R2

,�L= ��R1

+�R2� /2, and �L=�R1

. In addition, we assume�R2

=−�R1=3 , �d=�R1, and take for the period of the con-

sidered barrier modulation T=5. In a such case we integratenumerically the corresponding set of the differential equa-tions for the matrix elements of the evolution operator and inthe next step calculate the currents according to the formulaja�t�=−edna�t� /dt, where a=L , R1, or R2 and na�t� is givenin Eq. �5� �or similar to it�. We checked that the QD chargehardly depends on the additional VLR couplings. Althoughthe QD charge is almost insensitive to the additional overdotcouplings the currents demonstrate such dependence. Espe-cially visible are the differences for the case when the chemi-cal potential �L of the third electrode L lies between thechemical potentials of two other leads; see Figs. 11�B� and11�E�. For other values of �L, the influence of the overdottunneling channels for the parameters considered here issmaller. Note that after abrupt changing of the coupling be-tween the QD and R1 or R2 leads the currents jL, jR1

, and jR2rapidly change too, and after a short time reach the steadyvalues. The QD coupled with three leads could be consideredas the three-state system. We observe that for some values ofthe lead chemical potentials the currents change their valuesperiodically, e.g., from zero to the positive value �see jR2

in

FIG. 10. The averaged current derivatives d�jL�t�� /d�d against�d and � for x=0. We compare the results obtained for the QDcoupled with two leads—the upper �middle� panel—for �L

=5, �R=−8, �L=8, �d=4, �R=0 ��L=5,�R=0,�L=8,�d=4,�R

=2� with the results obtained for the QD coupled with three leads—the lower panel—�L=8, �d=4, �R1

=2 , �R2=0 , �L=5, �R1

=0 , �R2=−8, �=5.

FIG. 11. The time-dependent current flowing in the system of aQD coupled with the three leads: L, R1, and R2. The L lead iscoupled with the QD only—the left panels, and with the QD andtwo other leads, VLR1

=VLR2=4—the right panels. The couplings

between the QD and R1 , R2 leads are changed periodically. Theupper, middle, and lower panels correspond to �L=3, 0, and −3,respectively. −�R1

=�R2=3 , �d=0. The thin, thick, and broken

curves correspond to jL, jR1, and jR2

currents, respectively. The timeis in � /� units and currents are in e� /� units.


125312-10

Fig. 11�B��, from zero to the negative value �see jR1in Fig.

11�B�� or from the negative to the positive value �see jL inFig. 11�B��.The additional couplings between leads modifythe values of the currents but the qualitative picture remainsthe same. Note that in the first moment after abrupt changingof the coupling between the QD and R1 or R2 lead we havejL�t�+ jR1

�t�+ jR2�t��0 as in this case dnd�t� /dt�0 �not

shown here�. After some delay time the QD charge stabilizesaround its equilibrium value, the currents tend to constantvalues and their sum is equal to zero.

In the last step we consider the response of the currents tothe abrupt inclusion into the QD-2l system of the third elec-trode �in our case, L lead�. The results are presented in Fig.12 together with the schematic view of the subsequent tunnelconnections between the QD and the three leads. We showthe time-dependent currents jL�t�, jR1

�t�, and jR2�t� corre-

sponding to the three different ways of inclusion of the Llead. We assumed the chemical potentials �R1

=−�R2, �L

= ��R1+�R2

� /2, and �d=�R1. Consider the current jR2

flow-ing from the R2 lead characterized by the highest chemicalpotential �R2

. Before adding to the system of the L lead �seecase A in Fig. 12� the current jR2

has a constant value andflows from the R2 lead through the QD energy level to the R1lead. When the L lead is included into the system �the tun-neling coupling VLR1

changes abruptly at t=8 from the zeroto nonzero value� the current jR2

is almost unchanged—itsvalue decreases slightly without any transients at short timesafter the time t=8 �thin solid line�. Next, we consider thecase when the L lead is abruptly connected simultaneouslywith the QD and R1 lead �case B in Fig. 12�. Now the currentjR2

�thick solid line� decreases significantly during the short

time after the moment of inclusion of the L lead and settlesto its constant value. Note that jR2

decreases despite the ad-ditional charge transfer channel between R2 and L leads�through the QD energy level�. The destructive interferenceappears in this case as we have two transmission channels fortunneling electrons between R2 and R1 leads. However, theconstructive interference is visible if we consider the nextcase when the L lead is abruptly coupled with the QD-2lsystem assuming nonzero values of VLR1

, VLR2, and VLd �bro-

ken line�. Now, we have one additional charge-transfer chan-nel �from the R2 lead� in comparison with the former case.The current jR2

rapidly increases with some fluctuations andafter the time �� /� decreases to the constant value. A simi-lar analysis can be made considering the currents jL and jR1although the transient current changes are more visible nowat short times after the abrupt inclusion of additional electrontunneling channels. The above discussion concerns the spe-cific values of the lead chemical potentials and the positionof the QD energy level. Nevertheless, similar qualitative con-clusions can be made also for other values characterizing theconsidered system.

V. CONCLUSIONS

We have studied the time-dependent tunneling transportthrough the QD coupled with three metal leads using theevolution operator technique. The time-dependent QD chargeand currents were determined in terms of the appropriateevolution operator matrix elements. Applying the wide-bandlimit to the integrodifferential equations satisfied by the evo-lution operator matrix elements we were able to give theanalytical expressions for the time-averaged currents and dif-ferential conductance. We considered the external harmonicmicrowave fields applied to different parts of the consideredsystem, as well as the rectangular-pulse modulation imposedon different QD-lead barriers. In addition, we have studiedalso the time dependence of the currents due to abrupt inclu-sion into the QD–two leads system of the third electrode. Wehave considered also the effect of the additional couplingsbetween leads �we coupled one of the leads with the othertwo leads� on the conductance and current flowing in thesystem. Our main results can be summarized as follows:

�i� For the vanishing nonresonant tunneling path, VLR=0,and for the parameters for which the photon-assisted side-bands are clearly visible on the �jL�t�� curve the subsequentsideband peaks have the Lorentz-type form with the FWHMequal to 3� in comparison with 2� for the QD–two leadssystem. For the increasing value of VLR the form of the side-bands transforms from the Lorentz type to the fully asym-metric form for x=��13−�153� /8�0.28. For greater x theform of the sidebands changes and gains again the Lorentz-type shape for x=�2/2. For the QD coupled with two leadsthe corresponding values of x are equal to �3−2�2�0.41and 1, respectively.

�ii� For the vanishing VLR the differential conductancecurve, e.g., d�jL�t�� /d�L, possesses the sidebands of the Lor-entz type localized at �L=�d±�k. For VLR�0 these side-bands are described by the superposition of two parts, the

FIG. 12. �Color online� The time-dependent currents flowingfrom the L, R1, and R2 leads in the system shown in Fig. 1. The thinsolid, thick solid, and broken curves correspond to different cou-plings of the L lead to the other elements of the system: �VLR1=4 ,VLd=VLR2

=0�, �VLR1=4 ,VLd=4,VLR2

=0�, and �VLR1=VLd

=VLR2=4�, respectively.


125312-11

Lorentz-type and the asymmetric one centered at �d=�L±�k+2�x / �1+2x2� weighed by the factors ��1−13x2

+4x4� /�1+2x2�4�Jk2��d−�L� /�� and �4�x�1−2x2� / �1

+2x2�3�Jk2��d−�L� /��, respectively. For the QD–two leads

system the corresponding kth sidebands are centered at �d=�L±�k+�x / �1+x2� and their symmetric and asymmetricparts are weighted by the factors ��2�1−6x2+x4� /�1+x2�4�Jk

2��d−�L� /�� and �2�x�1−x2� /�1+x2�3�Jk2��d

−�L� /��, respectively.�iii� The symmetry properties of the sidebands corre-

sponding to the current flowing from the given lead dependon the position of the chemical potential of this lead in com-parison with the chemical potentials of the other two leads.Taking, for example, VLR=0 and �R2

localized in the middlebetween �L and �R1

one can observe on the �jL�t�� curve thesidebands of nearly regular �Lorentz-type� forms while thesidebands on the �jR2

�t�� curve have asymmetric structures.However, in the presence of the nonresonant tunneling paththe sidebands on the �jL�t�� curve change their form and forx�0.28 they have a fully asymmetric shape. On the otherhand, the sidebands on the �jR2

�t�� curve have for x�0.28 anearly Lorentz-like shape.

�iv� Especially large interference effects can be observedif we compare the current flowing in the QD-3l system withthe sum of currents flowing in the two QD-2l systems whichcan be viewed as the components of the considered morecomplicated QD-3l system. The difference between them isrelatively large independently of the overdot coupling be-tween leads �see Fig. 5�.

�v� In the case of strong asymmetry of the applied exter-nal field ��R1

�0,�L=�d=�R2=0� we observe for x=0 a

shoulder on the left side of the main resonant peak on the�jL�t�� curve vs the gate voltage. With the increasing overdotcoupling between L and R1 ,R2 leads the main resonant peakdisappears and transforms in a dip for strong coupling VLR.

�vi� Let us consider the time dependence of currentsflowing in response to the time-dependent barriers betweenthe QD and two leads �we assume a constant coupling of theQD with the third lead�. For the assumed rectangular-pulsemodulation applied to the QD–R1 and QD–R2 lead barriersone can consider the QD-3l system as a three-state one. Forexample, the currents jL�t�, jR1

�t�, and jR2�t� change their

values periodically between zero and positive, positive andzero, and negative and positive values, respectively �see Fig.11�B��. The additional couplings between the leads introduceonly small quantitative changes.

ACKNOWLEDGMENTS

The work of T.K. has been supported by the Foundationfor Polish Science and the work of R.T. has been partiallysupported by KBN Grant No. PBZ-MIN-008/P03/2003.

APPENDIX

In this section we present the derivations of the generalequations satisfied by the required functions Udd�t , t0�,Udk��t , t0�, Ud,r�/q��t , t0� and Uk�d�t , t0� , Uk�1,k�2

�t , t0� , Uk�,r�/q��t , t0�

needed for the calculations of the QD charge nd�t� and thecurrents flowing in the considered system. In the next step,using the WBL approximation we simplify these equationsand give the analytical solutions for them. Let us begin fromthe derivation of the integrodifferential equation satisfied byUdd�t , t0�. Writing down the formal solution of Eq. �9�,

Uq�/r�d�t,t0� = − i�t0

t

dt1Vq�/r�d�t1�Udd�t1,t0�

− i�t0

t

dt1�k�

Vq�/r�k��t1�Uk�d�t1,t0� �A1�

and inserting them to the formal solution of Eq. �8�,

Uk�d�t,t0� = − i�t0

t

dt1Vk�d�t1�Udd�t1,t0�

− i�t0

t

dt1 �p�=q� ,r�

Vk�p��t1�Up�d�t1,t0� , �A2�

one can obtain after straightforward calculations the functionUk�d�t , t0� expressed in the terms of Udd�t , t0�, only,

Uk�d�t,t0� = �− i��t0

t

dt1Vk�d�t1�Udd�t1,t0�

+ �− i�2�t0

t �t0

t1

dt1dt2Rk�d�t1,t2�Udd�t2,t0�

+ �j=2

�− i�2j�t0

t �t0

t1

¯ �t0

t2j−1

dt1 ¯ dt2j

� �k�1,k�2,…,k� j−1

Rk�k�1�t1,t2�Rk�2k�3

�t3,t4� ¯

Rk� j−1d�tj−1,tj�Udd�tj,t0�

+ �j=1

�− i�2j+1�t0

t �t0

t1

¯ �t0

t2j

dt1 ¯ dt2j+1

� �k�1,k�2,…,k� j

Rk�k�1�t1,t2�Rk�2k�3

�t3,t4� ¯

Vk� jd�t2j+1�Udd�t2j+1,t0� , �A3�

where

Rij�t1,t2� = �q�

Viq��t1�Vq� j�t2� + �r�

Vir��t1�Vr�j�t2� . �A4�

We remember that the wave vectors k�, q� , and r� correspond tothe left lead and the first and second right leads, respectively.Inserting into Eq. �7� the expressions for the functionsUq�d�t , t0�, Ur�d�t , t0�, and Uk�d�t , t0�, Eqs. �A1�–�A3�, we canwrite the integrodifferential equation for Udd�t , t0� in theform


125312-12


= − �t0

t

dt1 �k�

Vdk��t�Vk�d�t1� + Rdd�t,t0��Udd�t1,t0�

+ �j=1

�− i�2j−1�t0

t �t0

t1

¯ �t0

t2j−1

dt1 ¯ dt2j �k�1,k�2,…,k� j

�Rdk�1�t,t1�Rk�1k�2

�t2,t3� ¯

Rk� j−1k� j�t2j−2,t2j−1�Vk� jd

�t2j�

+ Vdk�1�t�Rk�1k�2

�t1,t2� ¯ Rk� jd�t2j−1,t2j��Udd�t2j,t0�

+ �j=1

�− i�2j�t0

t �t0

t1

¯ �t0

t2j

dt1 ¯ dt2j+1 �k�1k�2…k� j

�Rdk�1�t,t1�Rk�1k�2

�t2,t3� ¯

Rk� j−1k� j�t2j−2,t2j−1�Rk� jd

�t2j,t2j+1�

+ �k� j+1

Vdk�1�t�Rk�1k�2

�t1,t2� ¯ Rk� j,k�

j+1�t2j−1,t2j�Vk� j+1d�t2j+1��Udd�t2j+1,t0� . �A5�

This rather untractable general equation can be greatly sim-plified using the WBL approximation. Assuming Vdr�=Vdq�

=Vdk� �V , Vr�k� =Vq�k� �VLR, the multidimensional time integra-tions and summations over the wave vectors can be per-formed giving in result Eq. �12� with the solution

Udd�t,t0� = exp�− C1�t − t0�� . �A6�

Here C1= 32�−��3x2+2ix� / �1+2x2�� and x=VRL /D , D

being the bandwidth of the lead energy band �DR1=DR2

=DL=D�. The function Uk�d�t , t0� given in Eq. �A3� can bereduced within the WBL approximation to the form

Uk�d�t,t0� = − C2�t0

t

dt1Vk�d�t1�Udd�t1,t0� , �A7�

where

C2 =i + 2x

1 + 2x2 . �A8�

In order to calculate Udk��t , t0� we write down accordinglywith Eq. �3� the corresponding set of coupled differentialequations for the functions Udk��t , t0�, Uk�1k�2

�t , t0�, Uq�k��t , t0�,

and Ur�k��t , t0�. The subsequent steps of the calculations aresimilar to those performed in the derivation of Eq. �A5�.Inserting the formal solutions for Uk�1k�2

�t , t0� , Uq�/r�,k��t , t0�,

Uk�1k�2�t,t0� = �k�1k�2

− i�t0

t

dt1 Vk�1d�t1�Udk�2�t1,t0�

+ �p�=q�r�

Vk�1p��t1�Up�k�2�t1,t0�� , �A9�

Up�k��t,t0� = − i�t0

t

dt1 Vp�d�t1�Udk��t1,t0�

+ �k�1

Vp�k1�t1�Uk�1k��t1,t0�� , �A10�

into the differential equation satisfied by Udk��t , t0� one ob-tains the derivative �Udk��t , t0� /�t expressed in terms ofUdk��t , t0� and Uk�1k�2

�t , t0�. On the other hand, on the basis ofEqs. �A9� and �A10� the function Uk�1k�2

�t , t0� can be repre-sented in the form containing only Udk��t , t0�. Finally, oneobtains

�Udk��t,t0�

�t= − iVdk��t� + �− i�2�

t0

t

dt1 �q�1

Vdq�1Vq�1k��t1� + �

r�1

Vdr�1�t�Vr�1k��t1��

+ �− i�3�t0

t �t0

t1

dt1dt2 �k�1,q�1

Vdk�1�t�Vk�1q�1

�t1�Vq�1k��t2� + �k�1,r�1

¯�+ �− i�4�

t0

t �t0

t1 �t0

t2

dt1dt2dt3 �q�1,k�1,q�2

Vdq�1�t�Vq�1k�1

�t1�Vk�1q�2�t2�Vq�2k��t3� + �

r�1,k�1q�2

¯ + �q�1k�1r�2

¯ + �r�1k�1r�2

¯� + ¯

+ �− i�2�t0

t

dt1 �k�1

Vd�k�1�t�Vk�1d�t1� + �

q�1

Vdq�1�t�Vq�1d�t1� + �

r�1

Vdr�1�t�Vr�1d�t1��Udk��t1,t0�


125312-13

+ �− i�3�t0

t �t0

t1

dt1dt2 �k�1,q�1


�t1�Vq�1d�t2� + �k�1,r�1

¯ + �q�1k�1

¯ + �r�1k�1

¯�Udk��t2,t0� + ¯ . �A11�

In the WBL approximation this equation reduces to the form

�Udk��t,t0�

�t= − C2Vdk��t� + C3Udk��t,t0� , �A12�

with the solution

Udk�t,t0� = − C2�t0

t

dt1Vdk��t1�exp�− C3�t − t1�� , �A13�

where C3=��4ix−3� /2�1+2x2�. Next, taking into accountEqs. �A9� and �A10� the function Uk�1k�2

�t , t0� can be written interms of Udk�2

�t , t0�:

Uk�1k�2�t,t0� = �k�1k�2

− C2�t0

t

dt1Vk�1d�t1�Udk�2�t1,t0�

−2VLRx

1 + 2x2�t0

t

dt1ei��k�1−�k�2

��t1−t0�. �A14�

In order to calculate Udq��t , t0� we first write down thecoupled set of equations �on the basis of Eq. �3�� satisfied bythe functions Udq��t , t0�, Uq�1q�2

�t , t0�, Ur�q��t , t0�, and Uk�q��t , t0�.Inserting the formal solutions for the functions Uq�1q�2

�t , t0�,Ur�q��t , t0�, and Uk�q��t , t0� into the differential equation for thefunction Udq��t , t0� one obtains the derivative �Udq��t , t0� /�texpressed in terms of Udq�t , t0�, Uk�q��t , t0�, Uq�1q�2

�t , t0�, andUr�q��t , t0�. Inserting again into this equation the formal solu-tions for the functions Uq�1q�2

�t , t0�, Ur�q��t , t0�, and Uk�q��t , t0�,and repeating this process again and again, one obtains

�Udq��t,t0�

�t= − iVdq��t� + �− i�2�

t0

t

dt1�k�1

Vdk�1�t�Vk�1q��t1� + �− i�3�

t0

t �t0

t1

dt1dt2 �q�1,k�1


�t1�Vk�1q��t2� + �r�1k�1

¯�+ �− i�4�

t0

t �t0

t1 �t0

t2

dt1dt2dt3 �k�1q�1k�2


�t1�Vq�1k�2�t2�Vk�2q��t3� + �

k�1r�1k�2

¯ + ¯� + ¯

+ �− i�2�t0

t

dt1 �p�=k�1,q�1,r�1

Vdp��t�Vp�d�t1�Udq��t1,t0�

+ �− i�3�t0

t �t0

t1

dt1dt2 �q�1,k�1


�t1�Vk�1d�t2� + �r�1k�1

¯ + �k�1q�1

¯ + �k�1r�1

¯�Udq��t2,t0� + ¯ . �A15�

In the WBL approximation this equation is reduced to thefollowing form:

�Udq��t,t0�

�t= − C4Vdq��t� + C3Udq��t,t0� , �A16�

where C4= �1−x� / �1+2x2� and has the solution asfollows:

Udq��t,t0� = − C4�t0

t

dt1Vdq��t1�exp�− C3��t − t1�� . �A17�

The function Udr��t , t0� is identical with Udq��t , t0�.To calculate, e.g., the current jL�t�, we still need

the functions Uk�r��t , t0� and Uk�q��t , t0�. The functionUk�q��t , t0� can be obtained solving the set of thecoupled differential equations for the functions Udq��t , t0�,Uq�q�1

�t , t0�, Ur�q��t , t0�, and Uk�q��t , t0�. Writing downthe formal solution for Uk�q��t , t0� and insertinginto it, in the first step, the formal solutions forUq�1q�2

�t , t0� and Ur�q��t , t0� and, in the second step,the formal solutions for Udq��t , t0� and Uk�q��t , t0�,and so on, one obtains


125312-14

Uk�q��t,t0� = − i�t0

t

dt1Vk�d�t1�Udq�t1,t0�

+ �− i�2�t0

t

dt1dt2 �q�1

Vk�q�1�t1�Vq�1d�t2� + �

r�1

Vk�r�1�t�Vr�1d�t2��Udq��t1,t0�

+ �− i�3�t0

t �t0

t1 �t0

t2

dt1dt2dt3 �q�1,k�1

Vk�q1�t1�Vq�1k�1

�t2�Vk�1d�t3� + �r�1k�1

¯�Udq��t3,t0� + ¯

+ �− i��t0

t

dt2Vk�q��t1,t0� + �− i�3�t0

t �t0

t1 �t0

t2

dt1dt2d3 �q�1,k�1

Vk�q1�t1�Vq�1k�1

�t2�Vk�1d�t3� + �r�1k�1

¯� + ¯ . �A18�

Under the WBL approximation this equation reduces to the form

Uk�q��t,t0� = − C2�t0

t

dt1Vk�d�t1�Udq��t1,t0� −i

1 + 2x2�t0

t

dt1Vk�q��t1� , �A19�

where Udq�t , t0� is given in Eq. �A17�. The function Uk�r��t , t0� has the identical form as Uk�q��t , t0�.

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